Time-dependent Thermal Convection Research Articles

Numerical analysis of the potential for mixed thermal convection in the Buda Thermal Karst, Hungary

Study regionBuda Thermal Karst system, Hungary. Study focusThe pilot area has high geothermal potential characterized by prominent thermal anomalies, such as thermal springs and spas which tap the Triassic carbonate aquifers. Therefore, numerical simulations were carried out to examine the temperature field and flow pattern considering three successive heat transport mechanisms: thermal conduction, forced and mixed thermal convection in order to highlight the role of different driving forces of groundwater flow in the Buda Thermal Karst. New hydrological insights for the regionCompared to thermal conduction, topography-driven heat advection increases the surface heat flux. The superimposed effect of free thermal convection facilitates the formation of time-dependent mixed thermal convection from the deep carbonate layers. The Nusselt number varied between Nu = 1.56 and 5.25, while the recharge rate (R) ranged from R = 178 mm/yr to 250 mm/yr. Radiogenic heat production and hydraulically conductive faults have only a minor influence on the basin-scale temperature field and flow pattern. Boundary conditions prescribed on the temperature and pressure can considerably affect the numerical results. In each scenario, independently of the model parameters, time-dependent mixed thermal convection evolved both in the deep and the confined parts of the karstified carbonates of the Buda Thermal Karst system.

Numerical investigation of the combined effect of forced and free thermal convection in synthetic groundwater basins

The theoretical examination of the combined effect of water table configuration and heat transfer is relevant to improve understanding of deep groundwater systems, not only in siliciclastic sedimentary basins, but also in fractured rocks or karstified carbonates. Numerical model calculations have been carried out to investigate the interaction of topography-driven forced and buoyancy-driven free thermal convection in a synthetic, two-dimensional model. Effects of numerous model parameters were systematically studied in order to examine their influence on the Darcy flux, the temperature and the hydraulic head field. It was established that higher geothermal gradients and greater model depths facilitate the evolution of time-dependent free thermal convection in agreement with changes of the thermal Rayleigh number and the modified Péclet number. However, increasing water table slope and anisotropy coefficient favor the formation of stationary forced thermal convection. Free thermal convection mainly affects the deeper part of the midline and the discharge zone of the synthetic model. In the examined model basins, the position of the maximum hydraulic head is located within the bottom thermal boundary layer near the recharge zone. This divergent stagnation point underlies a local downwelling zone characterized by underpressure. These simulations draw attention to the importance of understanding the combined effect of forced and free thermal convection in sedimentary basins regarding regional groundwater flow patterns, and temperature distributions.

Reconstruction of time-dependent coefficients from heat moments

This paper investigates the inverse problems of simultaneous reconstruction of time-dependent thermal conductivity, convection or absorption coefficients in the parabolic heat equation governing transient heat and bio-heat thermal processes. Using initial and boundary conditions, as well as heat moments as over-determination conditions ensure that these inverse problems have a unique solution. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization. A discussion of the choice of multiple regularization parameters is provided. The finite-difference method with the Crank–Nicolson scheme is employed as a direct solver. The resulting inverse problems are recast as nonlinear minimization problems and are solved using the lsqnonlin routine from the MATLAB toolbox. Numerical results are presented and discussed.

Implicit–explicit schemes of finite element method for the non-stationary thermal convection problems with temperature-dependent coefficients

For the time-dependent thermal convection problems with temperature-dependent coefficients, the implicit–explicit scheme is presented, in which mixed finite element method is applied for the spatial approximation of the velocity, pressure and temperature while the time discretization is based on the high-order backward difference scheme. Linear terms are dealt with the implicit scheme while the nonlinear terms are treated by the semi-implicit scheme. The advantages for this scheme are unconditionally stable, decoupled computational and second order accuracy. Finally, numerical tests illustrate the theoretical results of the presented schemes, and display that the highly efficient method conserves the property of divergence free of the original problems.

Influences of the depth-dependence of thermal conductivity and expansivity on thermal convection with temperature-dependent viscosity

In this paper we carried out numerical experiments of a time-dependent thermal convection in a two-dimensional Cartesian box of aspect ratio (width/height) of 6, in order to study the influences on the convecting flow patterns of the spatial variations in physical properties (viscosity η, thermal conductivity k and expansivity α). A series of calculations by systematically varying the magnitude of spatial variations in these properties showed that the strongly temperature-dependent η induces the change in flow pattern into the “stagnant lid” (ST) regime, regardless of the increase in k and/or decrease in α with depth, where the convection occurs only beneath a stagnant lid of cold fluid at the top surface. In particular, we found that the increase in thermal conductivity k with depth significantly affects the convecting flow patterns in the presence of strong temperature-dependence in η. Compared with those with uniform k, the patterns of ST convection with non-uniform k are characterized by (i) thinner top thermal boundary layers or lids and (ii) larger horizontal length scales of convection cells beneath the stagnant lid. In addition, the variation in k with depth decreases the threshold values of the temperature-dependence in η above which the ST-mode of convection takes place, which may also help stabilize the convection cells of large horizontal length scales beneath stagnant lids. Our results may highlight the potential importance of the increase in thermal conductivity with depth (or pressure) in controlling the planforms of thermal convection in the mantle of terrestrial planets.

An improved formulation of the incompressible Navier–Stokes equations with variable viscosity

We present a new formulation of the incompressible Navier–Stokes equations with variable viscosity. By utilizing the incompressibility constraint to remove the trace from the deviatoric stress tensor, we eliminate second-order cross-derivatives of the velocity field, simplifying and improving the accuracy of co-located discretization techniques on both structured- and unstructured grids. This formulation improves the performance of SIMPLE-type algorithms that use sequential mass-momentum iterations to enforce incompressibility. A trace-free stress tensor also removes a typical source of net-rotation for simulations employing free-slip boundary conditions in spherical geometry. We implement the new scheme as a modification of an existing Boussinesq convection code, which we benchmark against analytical solutions of the Stokes problem in a spherical shell with both constant and radially dependent viscosity, and time-dependent thermal convection at infinite Prandtl number with large viscosity contrasts.

On the dynamics of 3-D single thermal plumes at various Prandtl numbers and Rayleigh numbers

Three-dimensional (3-D) numerical simulations of single turbulent thermal plumes in the Boussinesq approximation are used to understand more deeply the interaction of a plume with itself and its environment. In order to do so, we varied the Rayleigh and Prandtl numbers from Ra ∼ 105 to Ra ∼ 108 and from Pr ∼ 0.025 to Pr ∼ 70. We found that thermal dissipation takes place mostly on the border of the plume. Moreover, the rate of energy dissipation per unit mass ε T has a critical point around Pr ∼ 0.7. The reason is that at Pr greater than ∼0.7, buoyancy dominates inertia and thermal advection dominates wave formation whereas this trend is reversed at Pr less than ∼0.7. We also found that for large enough Prandtl number (Pr ∼ 70), the velocity field is mostly poloidal although this result was known for Rayleigh–Bénard convection (see Schmalzl et al. [On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 2004, 67, 390--396]). On the other hand, at small Prandtl numbers, the plume has a large helicity at large scale and a non-negligible toroidal part. Finally, as observed recently in details in weakly compressible turbulent thermal plume at Pr = 0.7 (see Plourde et al. [Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction. J. Fluid Mech. 2008, 604, 99--123]), we also noticed a two-time cycle in which there is entrainment of some of the external fluid to the plume, this process being most pronounced at the base of the plume. We explain this as a consequence of calculated Richardson number being unity at Pr = 0.7 when buoyancy balance inertia.

Three‐dimensional numerical simulation of g‐jitter induced convection and solute transport in magnetic fields

PurposeThe quality of crystals grown in space can be diversely affected by the melt flows induced by g‐jitter associated with a space vehicle. This paper presents a full three‐dimensional (3D) transient finite element analysis of the complex fluid flow and heat and mass transfer phenomena in a simplified Bridgman crystal growth configuration under the influence of g‐jitter perturbations and magnetic fields.Design/methodology/approachThe model development is based on the Galerkin finite element solution of the magnetohydrodynamic governing equations describing the thermal convection and heat and mass transfer in the melt. A physics‐based re‐numbering algorithm is used to make the formidable 3D simulations computationally feasible. Simulations are made using steady microgravity, synthetic and real g‐jitter data taken during a space flight.FindingsNumerical results show that g‐jitter drives a complex, 3D, time dependent thermal convection and that velocity spikes in response to real g‐jitter disturbances in space flights, resulting in irregular solute concentration distributions. An applied magnetic field provides an effective means to suppress the deleterious convection effects caused by g‐jitter. Based on the simulations with applied magnetic fields of various strengths and orientations, the magnetic field aligned with the thermal gradient provides an optimal damping effect, and the stronger magnetic field is more effective in suppressing the g‐jitter induced convection. While the convective flows and solute transport are complex and truly 3D, those in the symmetry plane parallel to the direction of g‐jitter are essentially two‐dimensional (2D), which may be approximated well by the widely used 2D models.Originality/valueThe physics‐based re‐numbering algorithm has made possible the large scale finite element computations for 3D g‐jitter flows in a magnetic field. The results indicate that an applied magnetic field can be effective in suppressing the g‐jitter driven flows and thus enhance the quality of crystals grown in space.

Transitions in thermal convection with strongly temperature-dependent viscosity in a wide box

Numerical models are systematically presented for time-dependent thermal convection of Newtonian fluid with strongly temperature-dependent viscosity in a two-dimensional rectangular box of aspect ratio 3 at various values of the Rayleigh number Ra b defined with viscosity at the bottom boundary up to 1.6×10 8 and the viscosity contrast across the box r η up to 10 8. We found that there are two different series of bifurcations that take place as r η increases. One series of bifurcations causes changes in the behavior of the thermal boundary layer along the surface boundary from small-viscosity-contrast (SVC) mode, through transitional (TR) mode, to stagnant-lid (ST) mode, or from SVC mode directly to ST mode, depending on Ra b. Another series of bifurcations causes changes in the aspect ratio of convection cells; convection with an elongated cell can take place at moderate r η (10 3–10 5.5 at Ra b=6×10 6), while only convection of aspect ratio close to 1 takes place at small r η and large r η. The parameter range of r η and Ra b for elongated-cell convection overlaps the parameter range for SVC and ST modes and include the entire parameter range for TR mode. In the elongated-ST regime, the lid of highly viscous fluid along the top boundary is not literally ‘stagnant’ but can horizontally move at a velocity high enough to induce a convection cell with aspect ratio much larger than 1.

A new convection–fractionation model for the evolution of the principal geochemical reservoirs of the Earth's mantle

There are geochemical reservoirs (CC, DM, PM, EM1, EM2, HIMU) in the Earth's mantle and crust. They are distinguished by their isotopic and chemical abundance ratios and they arise from the combination of partial melting, segregation, ascent of the melt, differentiation of the melt, and lateral transport. The fractionation generates the chemical and isotopic diversity, while the solid-state convection works toward homogenization in particular in mantle areas with high gradient of creeping velocity perpendicular to the velocity vector. The thermal and chemical evolution of the Earth's mantle and crust has been modeled simultaneously by a fractionation mechanism plus 2D-FD Oberbeck–Boussinesq thermal convection. In contrast with the published model K1 [Walzer, U., Hendel, R., 1997a. Time-dependent thermal convection, mantle differentiation and continental-crust growth. Geophys. J. Int. 130, 303–325; geological interpretation: Walzer, U., Hendel, R., 1997b. Tectonic episodicity and convective feedback mechanisms. Phys. Earth Planet. Interiors 100, 167–188], layered convection is not an assumption but the mineral phase changes are introduced for 410 and 660 km depth using customary values of the Clapeyron slope, the density contrast, etc. So we have heat sources and sinks at 410 and 660 km, respectively, in addition to the usual Rayleigh number, Rq, for internal heating by radioactivity and bottom heating at the core–mantle boundary (CMB). The viscosity is a function of the temperature field and the pressure. Segregation takes place if the asthenospheric viscosity falls below a certain threshold. Oceanic plateaus, enriched in incompatible elements, develop leaving behind depleted parts of the mantle (DM). The resulting inhomogeneous heat-source distribution generates a first feedback mechanism. A growing continent is produced by accretion and further fractionation, consuming the older oceanic plateaus. The lateral movability of the growing continent causes a second feedback mechanism. The mentioned mechanisms generate acceptable distributions of the convective vigor and of the growth of juvenile continent material over the time axis. These distributions are stable for a moderate variation of the parameters only. The solutions of the system of differential equations show acceptable values for the distributions of the temperature, viscosity, heat flow, mantle-creep velocity and the continent's velocity. The following new result is insensitive to a strong variation of Rq: After an initially rather complex mixing process of the depleted parts and the pristine parts of the mantle, we arrive at a mainly depleted upper half of the mantle including the uppermost parts of the lower mantle and a predominantly pristine lower half of the mantle in the Phanerozoic at the latest. There is no sharp interface between the halves.

Time-dependent thermal convection, mantle differentiation and continental-crust growth

SUMMARY The thermal evolution of the Earth is controlled by radioactive elements whose heat production rate decays with time and whose spatial distribution depends on chemical segregation processes. We present a 2-D and finite-difference Boussinesq convection model with temperature-dependent viscosity and time- and space-dependent radioactive heat sources. We used Newtonian rheology, boxes of aspect ratio 3, and heating from within. Starting from the geochemical results of Hofmann (1988), it is assumed that the radioactive heat sources of the mantle were initially distributed homogeneously. In a number of calculations, however, higher starting abundances of radioactive sources were assumed in the upper mantle. For the present geological situation, this also results in a depleted upper mantle. It was assumed that, if the viscosity falls below a certain critical value, chemical segregation will take place. In this way, model continental crust develops, leaving behind areas of a depleted mantle. We obtained the heat source, flow line, temperature, viscosity and heat-flow distribution as a function of time with realistic values, especially for the present time. The present viscosity of the upper mantle is approximately at the standard value obtained for postglacial uplift modelling; the deeper-mantle viscosity is considerably higher. The time dependence of the computed mean of the kinetic energy of mantle convection bears a resemblance to that of the magmatic and orogenetic activity of the Earth. We assumed that the 670 km discontinuity cannot be penetrated by the flow.

Ultrafast upwelling bursting through the upper mantle

A high-resolution calculation of strongly time-dependent thermal convection in the upper mantle with non-Newtonian, temperature-dependent rheology shows that, for an effective Rayleigh number of around 10 6, extremely fast upwellings, at times exceeding 10 m/yr, can be generated a few hundred kilometers below the lithosphere. There is a clear separation of timescales between this fast jet and the more slowly convecting mantle. Within this fast vertical shear layer is embedded a thermal boundary layer with a width of the order of 50 km. The development of the fast non-Newtonian upwelling is characterized by the growth of the plume head to a large enough size, before the plume takes off rapidly at a depth of around 350 km. Upon impinging the base of the lithosphere, this fast plume thins the lithosphere and the flow then becomes a horizontally moving hot sheet, extending out for around 1000 km. This scenario is found to repeat itself at the same location about 10 Myr after the first plume impingement.

Magnetic damping array implemented into a space-compatible multizone furnace for semiconductor crystal growth

For damping residual, unsteady flows in semiconductor melts a space-compatible multizone furnace including a magnetic damping array consisting of two radially magnetized permanent magnet rings has been developed. The performance of the whole system has been verified by measuring the temperature fluctuations caused by the time-dependent thermal convection in a Ga melt at 850 °C. By applying a static magnetic induction of 186 mT the fluctuations could be reduced from about 1.5 °C to less than 0.02 °C. For these high precision measurements optical fiber sensors have been used to measure the temperature in the Ga melt and to control the multizone furnace as well. A temperature constancy of ΔT≤0.03 °C in the melt could be established over a period of 16 h.

TRANSIENT SOLUTIONS BY A LEAST-SQUARES FINITE-ELEMENT METHOD AND JACOBI CONJUGATE GRADIENT TECHNIQUE

Abstract We present a least-squares finite-element method that can provide implicit, fully coupled transient solutions for time-dependent incompressible fluid flows and thermal convection. The algorithm consists of the Crank-Nicolson scheme for time discretization, Newton's method for linearization, and a matrix-free Jacobi conjugate gradient method as an iterative solver for the symmetric, positive-definite linear system of equations. The combined algorithm is first validated by two-dimensional flows: flows in a square cavity with a periodically oscillating lid and mixed convection in a driven cavity. Then the algorithm is used to obtain transient solutions of a three-dimensional lid-driven cavity flow for Re = 400

Comparison of steady-state and strongly chaotic thermal convection at high Rayleigh number.

Steady-state and time-dependent two-dimensional thermal convection in a Boussinesq, infinite-Prandtl-number fluid with stress-free boundaries has been investigated. Two independent numerical methods have been employed to calculate the evolution of convective flows in a rectangular box with aspect ratio \ensuremath{\lambda}=1.8 in a Rayleigh-number (Ra) range of ${10}^{6}$${10}^{9}$. With increasing Ra, greater than ${10}^{7}$, the flow reveals the presence of disconnected thermals, rather than connected plumes, driven by a persistent large-scale circulation. Such features have also been reported from laboratory convection experiments in the regime of hard turbulence. Extensive calculations were performed (up to 140 overturns) in order to reach the statistically stationary regime for strongly chaotic flows. A Gaussian distribution with a mean value ${\mathrm{Nu}}_{\mathit{t}}$ was derived from the time history of the Nusselt (Nu) numbers. The value of ${\mathrm{Nu}}_{\mathit{t}}$ can be directly obtained by solving the steady-state equations via an iteration procedure. Thus the stationary flow obtained from the steady-state method resembles the turbulent flow in a statistical sense. Since the iteration procedure is about ${10}^{4}$ times faster than calculating the full time-dependent evolution, it allows for the systematic investigation of the heat-transfer Nu-Ra relationship and other types of scaling laws. The steady-state and time-dependent experiments indicate that a power-law exponent of \ensuremath{\beta}=0.315 holds for the Nu-Ra relation for stress-free boundaries in the entire range of Ra. No indication of a jump in the exponent was found in the transition to hard turbulence.

Mass and heat transport in strongly time-dependent thermal convection at infinite prandtl number

Abstract We have studied heat and mass transport in two-dimensional, infinite Prandtl number, incompressible thermal convection for a range of Rayleigh numbers (Ra), between 106 and 108, and two different aspect-ratio boxes, between 1·8 and 10. This study has been motivated by recent developments in studying the transition from weak to strong turbulence in thermal convection. We have employed a two-dimensional finite element method in solving the time-dependent convection equations. Passive tracers, up to 900, have been put into the flow fields for monitoring the style and pattern of mass transport. At Ra around 106 convection does not take place in a strictly cellular mode. Thermals are ejected from the hot and cold boundary layers. These boundary-layer instabilities enhance the mass transport in the interior of individual cells. The fate of these instabilities is determined ultimately by the large-scale circulation. The persistent large-scale circulation gives rise to a significant decrease of the heat ...

Time-dependent large aspect-ratio thermal convection in the earth's mantle

Abstract Numerical simulations of two-dimensional time-dependent thermal convection in a Boussinesq, isoviscous, infinite Prandtl number fluid with isothermal, stress-free boundaries have been performed in large aspect-ratio configurations, in which the fluid is heated from below as well as internally. The value of the basal heated Rayleigh number ranged from 16000 to 800 000 and the Rayleigh number based on internal heat generation was varied from zero to 4 500 000. Large aspect-ratio cells are found to exist, however, they are time-dependent even at small values of the Rayleigh number. In the absence of internal heating, the onset of time-dependence occurs as a regular oscillation in the flow characteristics (Nusselt number, kinetic energy), and is accompanied by the presence of boundary layer instabilities (BLI) which exist within a large aspect-ratio circulation. At high values of the Rayleigh number the BLI are powerful features which leave the confines of the boundary layer and strongly perturb the large aspect-ratio circulation giving the flow a multi-scale character. Convective mixing of these powerful BLI results in heterogeneity in the cell interior which plays a role in the excitation of new BLI and establishes a negative temperature gradient in the cell interior. We have developed a fluid loop model which gives a qualitative explanation for the variation of the onset of time-dependence with aspect-ratio. The addition of internal heating tends to destabilize the large aspect-ratio cell configuration. Multi-cellular states last longer and occur more frequently with increasing amounts of internal heating. These calculations shed new light on a variety of time-dependent phenomena in geodynamics such as subduction, back-arc spreading, intraplate deformation, and the average geotherm. Recently, Jeanloz and Morris proposed that the seismic inhomogeneity parameter (η) can be used to measure the importance of internal heating in mantle convection. However, our calculations indicate that η cannot be used to gauge the importance of internal heating in mantle convection.

Instabilities of Steady, Periodic, and Quasi-Periodic Modes of Convection in Porous Media

Instabilities of steady and time-dependent thermal convection in a fluid-saturated porous medium heated from below have been studied using linear perturbation theory. The stability of steady-state solutions of the governing equations (obtained numerically) has been analyzed by evaluating the eigenvalues of the linearized system of equations describing the temporal behavior of infinitesimal perturbations. Using this procedure, we have found that time-dependent convection in a square cell sets in at Rayleigh number Ra=390. The temporal frequency of the simply periodic (P(1)) convection at Rayleigh numbers exceeding this value is given by the imaginary part of the complex eigenvalue. The stability of this (P(1)) state has also been studied; transition to quasi-periodic convection (QP2) occurs at Ra ≈ 510. A reverse transition to a simply periodic state (P(2)) occurs at Ra ≈ 560; a slight jump in the frequency of the P(2) state occurs at Ra between 625 and 640. The jump coincides with a second narrow (in terms of Ra) region of quasi-periodicity.

Time-dependent thermal convection in a stably stratified fluid layer heated from below

Time-dependent thermal convection resulting when a stably stratified fluid layer is heated from below is analyzed as an initial-value problem of a linear model. The effect of stratification on its development and structure is investigated. The convection is found to develop in three stages. Initially the given disturbance induces oscillatory motion with the corresponding Brunt–Väisälä frequency. In the second stage the stratification works to form secondary cells above the primary cell. This multiple cell pattern, however, changes slowly but periodically with time. Finally, the convection grows with an exponential growth rate, retaining a similar cell pattern. The vertical scale of the primary cell decreases with the stratification level, while the horizontal scale remains constant. The onset time of the manifest convection is predicted to be delayed as the stratification shifts from neutral to stable conditions. Attaining a maximum, it then becomes slightly earlier. Experimental results supporting the model predictions are also presented.

Transitions in time-dependent thermal convection in fluid-saturated porous media

Numerical simulations of single-cell, two-dimensional, time-dependent thermal convection in a square cross-section of fluid-saturated porous material heated uniformly from below reveal a series of transitions between distinct oscillatory dynamical regimes. With increasing Rayleigh number R, the flow first evolves from steady-state behaviour into periodic motion with a single frequency f which depends on R approximately according to . The two frequencies in the narrow transition regime may be locked to a rational ratio, in which case the flow is periodic, or they may be commensurate, in which case the flow is quasi-periodic. The spectral characteristics of numerical realizations of unsteady convection and the occurrences of transitions therein are highly dependent on truncation level in Galerkin schemes or resolution in finite-difference approaches.